CN108629140A - A kind of composite structure design optimization method with holes based on geodesic curve distance - Google Patents
A kind of composite structure design optimization method with holes based on geodesic curve distance Download PDFInfo
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Abstract
The invention belongs to composite structure design optimization fields, and specifically disclose a kind of composite structure design optimization method with holes based on geodesic curve distance, include the following steps：Discrete design point is uniformly defined in the entity area of composite structure with holes, sets the initial value of fiber angles at each design point；Mesh generation is carried out to structure, determines the coordinate of each unit central point；Each design point is calculated to the geodesic curve distance of each unit central point, the fiber angles of each unit center are acquired based on geodesic curve distance；Stiffness matrix is established, and global displacement vector is calculated；Sensitivity is obtained according to global displacement vector, then updates fiber angles θiUntil meeting optimization end condition.The present invention passes through computing unit central point to the geodesic curve distance between design point, and the continuous varied angle fiber layout of composite structure with holes is obtained based on geodesic curve distance, the continuous varied angle fiber layout for improving composite structure with holes, improves structural behaviour.
Description
Technical field
The invention belongs to composite structure design optimization field, more particularly, to a kind of based on geodesic curve distance
Composite structure design optimization method with holes.
Background technology
With it is conventional determine rigidity composite structure compared with, wavy fiber composite construction has high designability.It removes
Conventional construction profile design is outer, and the design of composite structure further includes the design that fiber is laid out.For example, fine by designing
The lay angle of dimension makes structure have the characteristic of variation rigidity, and then obtains more preferably structural behaviour.With automatic fibers lay and
The development of increases material manufacturing technology, wavy fiber composite construction are gradually applied to the fields such as aerospace, automobile, high-end equipment.
For the design that fiber angles in composite structure are laid out, due to the spatial continuity at fleece-laying angle to be ensured
In order to fabricate, so continuous varied angle fiber layout descriptive model is essential.Current existing optimization design
In method, the fiber-reinforced composite materials structures optimization method based on Shepard interpolation can guarantee that the space of fiber angles is continuous
Variation layout, such as a kind of fiber-reinforced composite materials structures optimization based on Shepard interpolation disclosed in CN107590325A
Method, a kind of wavy fiber composite structural design Waterfall type multistage based on Shepard interpolation disclosed in CN107729648A are excellent
Change method.But the Shepard interpolation that the above method is related to is based on Euclidean distance, can only solve the composite structure without hole
Design, and composite structure design problem with holes is not considered.In actual engineering structure, most of composite structure
Band hole, or be non-convex shape.To solve the fiber angles optimization problem of such composite structure, research and design one is needed
The new optimization method of kind, to realize the optimization of the fiber angles with hole composite structure.
Invention content
For the disadvantages described above or Improvement requirement of the prior art, the present invention provides a kind of based on the with holes of geodesic curve distance
Composite structure design optimization method by computing unit central point to the geodesic curve distance between design point, and is based on
Geodesic curve distance obtains the continuous varied angle fiber layout of composite structure with holes, solves existing based on Euclidean distance
The limitation of Shepard interpolation methods improves the continuous varied angle fiber layout of composite structure with holes, improves structure
Performance.
To achieve the above object, it is excellent to propose a kind of composite structure with holes design based on geodesic curve distance by the present invention
Change method comprising following steps：
S1 uniformly defines series of discrete design point p in the entity area of composite structure with holesi, and set and respectively set
Enumeration piLocate fiber angles θiInitial value, wherein i=1,2 ..., n, n are positive integer；
S2 carries out mesh generation to composite structure with holes, extracts node data and cell data, determines in each unit
Heart point xcCoordinate；
S3 calculates each unit central point xcTo each design point piGeodesic curve distance | | xc-pi||g, it is based on geodesic curve distance | | xc-pi|
|gAcquire each unit central point xcThe fiber angles at placeWherein,
IxcFor unit center point xcThe domain of influence in all design points index set；
S4, which is established, depends on fiber angles θeStiffness matrix, and be calculated global displacement vector；
S5 obtains sensitivity according to global displacement vector, then updates fiber angles θiUntil meet optimization end condition, with
This completes the design optimization of composite structure with holes.
As it is further preferred that geodesic curve distance | | xc-pi||gIt obtains in the following way：
1) background grid of square is divided in the entity area of composite structure with holes, defines velocity field F (x),
According to nonlinear partial differential equationSolve each unit central point xcGeodetic between corresponding background grid node x
Linear distance d；
2) each unit central point x for utilizing two-dimensional linear interpolation to be calculated according to step 1)cWith corresponding background grid section
Geodesic curve distance d computing unit central points x between point xcWith design point piBetween geodesic curve distance | | xc-pi||g。
As it is further preferred that velocity field F (x) is specially：
Wherein, x is the coordinate of background grid node, and Ω represents the entity area of composite structure with holes, and x ∈ Ω are indicated
Node in entity area,Indicate node in hole.
As it is further preferred that geodesic curve distance | | xc-pi||gSpecifically calculated using following formula：
Wherein,
It is indicated respectively from unit center point xcTo four background grid node A1、A2、A3、A4Geodesic curve distance, this four background grids
Node is respectively distributed to design point piUpper left, lower-left, upper right, lower right, (a1,b1) it is A1Coordinate, (a1,b2) it is A2's
Coordinate, (a2,b1) it is A3Coordinate, (a2,b2) it is A4Coordinate, (a, b) be piCoordinate.
As it is further preferred that update fiber angles θiUntil satisfaction optimization end condition is specially：
(1) update fiber angles θi：
Wherein,For+1 updated θ of kthiValue,For the updated θ of kth timeiValue, η is step factor,
For sensitivity valueUsing after Conjugate Mapping as a result,WithVariable update value under respectively step-length constraint δ；
(2) according to updated fiber angles calculating target function value, judge whether target function value meets the condition of convergence,
If it is not, return to step (1) updates fiber angles again, if so, terminating, corresponding fiber angles are optimum results at this time,
The design optimization of composite structure with holes is completed with this.
As it is further preferred that the condition of convergence is：
Wherein, ckFor the updated target function value of kth time, ck-1For -1 updated target function value of kth, k is fibre
Tie up the newer number of angle.
As it is further preferred thatWithIt is determined using following formula：
Wherein, θminAnd θminRespectively θiBound, θmin=0, θmax=θmin+ π-ε, ε=1 × 10-8。
In general, through the invention it is contemplated above technical scheme is compared with the prior art, mainly have below
Technological merit：
1. the present invention changes existing optimization method, by computing unit central point to the geodesic curve distance between design point,
And the continuous varied angle fiber layout of composite structure with holes is obtained based on geodesic curve distance, it solves existing based on Euclidean
The limitation of the Shepard interpolation methods of distance improves the continuous varied angle fiber layout of composite structure with holes, is promoted
Structural behaviour.
2. due to the presence of composite structure hole with holes so that the existing optimization method based on Euclidean distance does not exist
Suitable for the structure, because existing method ignores at the unit center point of hole both sides fiber angle at fiber angles and design point
The irrelevance of degree leads to the fiber irrational distribution of hole both sides and keeps the rigidity of structure of design less than normal, and the present invention is by grinding
Study carefully and obtain a kind of new optimization method, is applicable to the design optimization of composite structure with holes, this method is compared to existing
Method can greatly promote the rigidity of structure of design, improve about 30%, and the fiber angles distribution around hole is more reasonable.
3. the method for the present invention is by dividing background grid and defines velocity field, to distinguish entity area and perforated, so
Calculating design point in each region based on above-mentioned differentiation correspondence afterwards can preferably solve to the geodesic curve distance of unit center point
The fiber angles design problem of composite structure with holes improves the fiber distribution around hole, and the mechanical property of lift structure can
Obtain optimal continuous varied angle fiber layout.
Description of the drawings
Fig. 1 is the flow chart of the composite structure design optimization method with holes based on geodesic curve distance of the present invention；
Fig. 2 is the plane cantilever beam structure optimization design example schematic diagram that better embodiment of the present invention provides；
Fig. 3 is optimum results of the optimization design example about the fiber angles at design point in Fig. 2；
Fig. 4 is the optimum results of optimization design example in Fig. 2 about the fiber angles of finite elements center；
Fig. 5 is the optimum results of the finite elements center fiber angles obtained based on Euclidean distance optimization method.
Specific implementation mode
In order to make the purpose , technical scheme and advantage of the present invention be clearer, with reference to the accompanying drawings and embodiments, right
The present invention is further elaborated.It should be appreciated that the specific embodiments described herein are merely illustrative of the present invention, and
It is not used in the restriction present invention.As long as in addition, technical characteristic involved in the various embodiments of the present invention described below
It does not constitute a conflict with each other and can be combined with each other.
As shown in Figure 1, a kind of composite structure with holes design based on geodesic curve distance provided in an embodiment of the present invention
Optimization method includes the following steps：
S1 uniformly defines series of discrete design point p in the entity area of composite structure with holesi, and set and respectively set
Enumeration piLocate fiber angles θiInitial value, wherein i=1,2 ..., n, n are positive integer；
S2 carries out mesh generation to composite structure with holes, extracts node data and cell data, determines in each unit
Heart point xcCoordinate；
S3 calculates each unit central point xcTo each design point piGeodesic curve distance | | xc-pi||g, it is based on geodesic curve distance | | xc-pi|
|gAcquire each unit central point xcThe fiber angles at placeWherein,
IxcFor unit center point xcThe domain of influence in all design points index set；
S4, which is established, depends on fiber angles θeStiffness matrix, and be calculated global displacement vector；
S5 obtains sensitivity according to global displacement vector, then updates fiber angles θiUntil meet optimization end condition, with
This completes the design optimization of composite structure with holes.
Below by way of specific embodiment, the present invention is described in detail, in the present embodiment, with outstanding with uniform load
The present invention is explained for the optimization problem that arm beam flexibility minimizes.As shown in Fig. 2, the setting in given 1m × 2m design domains
Fiber angles initial value be 90 °, region left border apply displacement constraint, perpendicular to beam direction loads in plane f=1 equably
It is applied on the right side of the coboundary of region within the scope of 0.5m, it is excellent to carry out fiber angles layout to fiber reinforcement cantilever beam structure with holes
Change, so that its flexibility is minimized, include the following steps：
S1 uniformly defines series of discrete design point p in the entity area of composite structure with holesi, and set and respectively set
Enumeration piLocate fiber angles θiInitial value, wherein i=1,2 ..., 200, design point is uniformly arranged, as shown in figure 3, being designed as
10 × 20, give piLocate fiber angles θiInitial value be 90 °, i.e. fiber and horizontal angle is 90 °；
S2 carries out mesh generation to composite structure with holes, extracts node data and cell data, determines in each unit
Heart point xcCoordinate, number of nodes 807, unit number NeIt is 700；
Specifically, as shown in figure 4, quadrilateral units can be selected to draw hole structure in ANSYS finite element analysis softwares
Divide finite element grid, extracts node data and cell data, wherein node data includes node serial number, node coordinate, unit number
It is average according to the abscissa of each unit corresponding node according to the number for including corresponding four nodes of element number and each unit
Value and ordinate average value, you can obtain each unit central point xcTransverse and longitudinal coordinate, each unit central point xcAbscissa be equal to should
The average value of the abscissa of the corresponding all nodes of unit, each unit central point xcOrdinate to be equal to the unit corresponding all
The average value of the ordinate of node, wherein can be arbitrary to be laid in plane using the laying plane of fiber as two-dimensional coordinate plane
It is some coordinate origin, using horizontal line from left to right as X-axis, using the horizontal line vertical with X-axis from front to back as Y-axis, tool
Body can be designed according to actual needs.
S3 calculates each unit central point xcTo each design point piGeodesic curve distance | | xc-pi||g, it is based on geodesic curve distance | | xc-pi|
|gAcquire each unit central point xcThe fiber angles at placeWherein,
IxcFor unit center point xcThe domain of influence in all design points index set, influence domain radius RcSelection to ensure at least one
A design point piHeart point x in the cellscThe domain of influence in, preferred Rc=0.3 to meet the requirements.Due to unit center point xcWith
Design point piPosition coordinates fix, the geodesic curve distance being calculated before Optimized Iterative | | x-pi||gWith weight function wi(x)
It is saved, to save the optimization time.
Specifically, geodesic curve distance | | xc-pi||gIt calculates and obtains in the following way：
The background grid of square, each vertex of square net are divided in the entity area of composite structure with holes
Referred to as node, x are the coordinate of background grid node, and x ∈ Ω indicate node in entity area,Indicate node in hole
It is interior, the wherein entity area of Ω representative structures；Under the grid background, velocity field F (x) is defined, is indicated as follows：
Eikonal nonlinear partial differential equations can be used between F (x) and d to be indicated：
Wherein, d is indicated from unit center point xcGeodesic curve distance between background grid node x,Indicate gradient,
| | euclideam norm is indicated, as unit center point xcD=0 when being overlapped with background grid node x, it is non-linear inclined by solving
The differential equation, which can solve, obtains geodesic curve distance d, and for the method for solving of nonlinear partial differential equation, it can adopt
It is solved with existing conventional method for solving, such as is solved using fast-marching algorithm, be the prior art, do not go to live in the household of one's in-laws on getting married herein
It states, the specific method for solving present invention does not limit, and only needs that parameter to be solved in nonlinear partial differential equation can be solved.
For example, unit center point xcFor (1,0.65), background grid node x is (1,0.35), is distributed in entity area, then F (x)=
1, solve equation using fast-marching algorithmObtain d=1.175；For another example, unit center point xcFor (1,0.65),
Background grid node x is (1,0.5), is distributed in perforated, then F (x)=1 × 10-8, equation is solved using fast-marching algorithmObtain d=2.684 × 106。
Utilize two-dimensional linear interpolation calculation unit center point xcWith design point piBetween geodesic curve distance | | xc-pi||g, tool
Body calculates as follows：
Wherein, d1、d2、d3、d4It is indicated respectively from unit center point xcTo four background grid node A1、A2、A3、A4Survey
Ground linear distance is obtained by solving nonlinear partial differential equation, this four background grid nodes are respectively distributed to design point pi
Upper left, lower-left, upper right, bottom-right location, such as design point piIt is distributed in background grid, then four nodes of the background grid
As this four nodes, for another example design point piIt is distributed in background grid, then design point piUpper left, lower-left, upper right, bottom right position
Set corresponding and design point piFour adjacent nodes are this four nodes, wherein A1Coordinate be (a1,b1)、A2Coordinate
For (a1,b2)、A3Coordinate be (a2,b1)、A4Coordinate be (a2,b2), piCoordinate be (a, b).
S4, which is established, depends on fiber angles θeStiffness matrix, and be calculated global displacement vector；
S5 obtains sensitivity according to global displacement vector, then updates fiber angles θiUntil meeting optimization end condition.
Wherein, the computational methods one for the existing patent introduced in step S4 and S5 is used computational methods and background technology
There are essential distinctions for the initial data for causing, but inputting, and initial data of the invention is geodesic curve distance, and existing patent is original
Data are Euclidean distances, therefore data processing details is distinct.
Specifically, in step S4, the finite element grid divided using step S2, unit number NeIt is 700, in each unit
It is established dependent on fiber angles θ at unit center on e (e=1,2,3 ..., 700)eStiffness matrix Ke(θe), wherein：
In formula, B is displacement strain matrix, D (θe) it is to rely on the unitary elasticity matrix of fiber angles at unit center, A
For the area of structure, AeFor the area of unit e；
Unitary elasticity matrix D (θe) calculate it is as follows：
D(θe)=T (θe)D0T(θe)T
In formula, D0Original elastic matrix when not rotated for fiber, T (θe) it is spin matrix：
Wherein, ExAnd EyIt is Young's modulus, GxyIt is modulus of shearing, νxyAnd νyxIt is Poisson's ratio, meets condition νxyEy=νyxEx；
Pass through module units stiffness matrix K againe(θe) Bulk stiffness matrix K is obtained, concrete operations are exactly all units
Stiffness matrix Ke(θe) expand as the square battle array of Bulk stiffness matrix same orderThe element of element stiffness matrix is stored
The row and column integrally numbered to corresponding node in cell inflation matrix, remaining element is zero, and the matrix after expansion is also referred to as unit
Matrix is contributed, all units contribution matrix is added to get to Bulk stiffness matrix, calculation formula is as follows：
Global displacement vector u is calculated according to Ku=f, wherein f is outer force vector, can be selected according to actual needs
It selects and sets.
In step S5, stiffness matrix K is utilizede(θe)、θeAnd θeWith θiRelation derivation object function c about design variable
θiSensitivitySensitivityCalculation formula is as follows：
In formula, ueNumber for the corresponding motion vectors of unit e, unit e is ne, then ueFor global displacement vector u n-theRow
Element.
Utilize the optimization algorithm update design variable θ based on sensitivity Conjugate MappingiUntil convergence, more format is as follows：
Wherein,For+1 updated fiber angle angle value of kth,For the updated fiber angle angle value of kth time, k is
Newer number is >=1 positive integer,For initial fiber angle, i.e. the initial value of the fiber angles set in step S1, η
For step factor, value 3；
For sensitivity valueAfter normalization using after Conjugate Mapping as a result, calculation formula is as follows：
Wherein, e is natural constant；
WithVariable update value under respectively step-length constraint δ (value is 5 °), is defined as follows：
Wherein, θminAnd θminRespectively θiBound, θmin=0, θmax=θmin+ π-ε, ε=1 × 10-8, it is one
Minimum positive value, numerical fault caused by being calculated due to trigonometric function when for eliminating interpolation；
Obtain updated design variable (i.e. fiber angles), such as+1 updated θ of kthiValue isThen lead to
It crosses finite element analysis and obtains+1 updated global displacement vector u of design variable kthk+1, utilize formula ck+1=fTuk+1It obtains
ck+1, until meeting following convergence criterion, complete entire optimization process：
Any one condition in above three condition ought be met and complete optimization, wherein ckIt is updated for kth time
Target function value, ck-1For -1 updated target function value of kth, k is the newer number of fiber angles.
To sum up, the mathematical optimization models of composite structure with holes may be summarized to be：
findθi(i=1,2 ..., 200)
Min c=fTu
S.t.Ku=f
θmin≤θi≤θmax
Wherein, the fiber angle angle value θ at discrete design pointiFor design variable, object function is flexibility c, and design object makes
The flexibility c of structure is minimized, and design constraint is equilibrium equation Ku=f and θiBound θmax、θmin。
The optimum results of the present invention are as follows：Subtle angle at composite structure design point with holes based on geodesic curve distance
Optimum results are spent as shown in figure 3, the fiber angles optimum results of finite elements center are as shown in figure 4, corresponding structure is soft
Angle value is 105.29；And composite structure with holes is optimized based on the optimization method of Euclidean distance using existing,
Finite elements center fiber angles optimum results as shown in figure 5, corresponding structural compliance value be 151.64.Compare this
Invention and existing optimization method, it can be seen that the rigidity of structure bigger (flexibility smaller) that the method for the present invention obtains about improves
30%, and the fiber angles distribution around hole is more reasonable, therefore the method for the present invention is more suitable for composite structure with holes
Design optimization.
As it will be easily appreciated by one skilled in the art that the foregoing is merely illustrative of the preferred embodiments of the present invention, not to
The limitation present invention, all within the spirits and principles of the present invention made by all any modification, equivalent and improvement etc., should all include
Within protection scope of the present invention.
Claims (7)
1. a kind of composite structure design optimization method with holes based on geodesic curve distance, which is characterized in that including walking as follows
Suddenly：
S1 uniformly defines series of discrete design point p in the entity area of composite structure with holesi, and set each design point
piLocate fiber angles θiInitial value, wherein i=1,2 ..., n, n are positive integer；
S2 carries out mesh generation to composite structure with holes, extracts node data and cell data, determines each unit central point xc
Coordinate；
S3 calculates each unit central point xcTo each design point piGeodesic curve distance | | xc-pi||g, it is based on geodesic curve distance | | xc-pi||g
Acquire each unit central point xcThe fiber angles at placeWherein,
IxcFor unit center point xcThe domain of influence in all design points index set；
S4, which is established, depends on fiber angles θeStiffness matrix, and be calculated global displacement vector；
S5 obtains sensitivity according to global displacement vector, then updates fiber angles θiUntil meeting optimization end condition, completed with this
The design optimization of composite structure with holes.
2. the composite structure design optimization method with holes based on geodesic curve distance, feature exist as described in claim 1
In geodesic curve distance | | xc-pi||gIt obtains in the following way：
1) background grid of square is divided in the entity area of composite structure with holes, defines velocity field F (x), according to
Nonlinear partial differential equationSolve each unit central point xcGeodesic curve between corresponding background grid node x away from
From d；
2) each unit central point x for utilizing two-dimensional linear interpolation to be calculated according to step 1)cBetween corresponding background grid node x
Geodesic curve distance d computing unit central points xcWith design point piBetween geodesic curve distance | | xc-pi||g。
3. the composite structure design optimization method with holes based on geodesic curve distance, feature exist as claimed in claim 2
In velocity field F (x) is specially：
Wherein, x is the coordinate of background grid node, and Ω represents the entity area of composite structure with holes, and x ∈ Ω indicate node
In entity area,Indicate node in hole.
4. the composite structure design optimization method with holes based on geodesic curve distance as claimed in claim 2 or claim 3, feature
It is, geodesic curve distance | | xc-pi||gSpecifically calculated using following formula：
Wherein,d1、d2、d3、d4Indicate respectively from
Unit center point xcTo four background grid node A1、A2、A3、A4Geodesic curve distance, this four background grid nodes divide respectively
Cloth is in design point piUpper left, lower-left, upper right, lower right, (a1,b1) it is A1Coordinate, (a1,b2) it is A2Coordinate, (a2,
b1) it is A3Coordinate, (a2,b2) it is A4Coordinate, (a, b) be piCoordinate.
5. the composite structure design optimization method with holes based on geodesic curve distance, feature exist as described in claim 1
In update fiber angles θiUntil satisfaction optimization end condition is specially：
(1) update fiber angles θi：
Wherein,For+1 updated θ of kthiValue,For the updated θ of kth timeiValue, η is step factor,It is quick
Angle valueUsing after Conjugate Mapping as a result,WithVariable update value under respectively step-length constraint δ；
(2) according to updated fiber angles calculating target function value, judge whether target function value meets the condition of convergence, if
No, return to step (1) updates fiber angles again, if so, terminating, corresponding fiber angles are optimum results at this time, with
This completes the design optimization of composite structure with holes.
6. the composite structure design optimization method with holes based on geodesic curve distance, feature exist as claimed in claim 5
In the condition of convergence is：
|ck-ck-1|≤2 orOr k ＞ 50
Wherein, ckFor the updated target function value of kth time, ck-1For -1 updated target function value of kth, k is fiber angle
Spend newer number.
7. the composite structure design optimization method with holes as claimed in any one of claims 1 to 6 based on geodesic curve distance,
It is characterized in that,WithIt is determined using following formula：
Wherein, θminAnd θminRespectively θiBound, θmin=0, θmax=θmin+ π-ε, ε=1 × 10-8。
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